Building Kepler's platonic-solid Solar System

 kw: book reviews, science fiction, space fiction, generation starships, solar system engineering, totalitarianism, culture shock

It takes a lot of backstory to assemble the worldviews in Fortress Sol by Stephen Baxter. Rather than drive off readers with 10-plus chapters of stage-setting (in a 37-chapter book), the author wisely metes it out in digestible portions as needed.

Seen from afar—say, half a light-year away in the Oort cloud—two Earth-born cultures, separated by a millennium, are about to collide.

• The generation starship Lightbird has been traveling for a century from Ross 128b, a habitable planet of the red dwarf star Ross 128, eleven light-years from the Solar System. Energy for its propulsion is from lumes, enigmatic creatures of deep space that can apparently convert matter to antimatter, or extract antimatter from vacuum energy. The author doesn't dig too far, a decision I support. Ross 128b was colonized a thousand years earlier.
• The residents of the Solar System spent a good part of that thousand years enveloping a sphere a little larger than the orbit of Uranus in the Mask, a carbon nanotube structure that hides the inner seven planets from the universe, and hides the universe from within. Material for the Mask, and for an inner structure, the Wrap that hides the Sun from view, was derived from partial demolition of Jupiter and the other giant planets, except for Neptune, which was destroyed by lumes, an event that started the whole adventure. The lumes were thought to be weapons of a super-advanced interstellar culture that was preparing to invade. A millennium later, no invasion has yet arisen. An interplanetary "railway" called the Frame was also built to facilitate travel everywhere inside the Mask, but mostly in the orbital plane of the seven surviving planets.

Both cultures are totalitarian in nature. Aboard Lightbird, everything must contribute to the progress of the four-generation mission of reaching the Solar System. That includes decisions about who is allowed to be born, and such a decision is at the core of a neurosis that afflicts a young woman named Muree. She is one of a few who care for the onboard lume tank. The lumes have proven useful, and an ecology that promotes a tank containing about 100 of them is managed such that they create a new lume about each day, even as another lume is consumed in providing energy to drive the constant acceleration of the ship, and power its life support mechanisms. (Note, the amount of antimatter needed for this would be just a few grams per day, so a different consumption rate might be warranted. I don't know how much a lume is supposed to weigh.)

Ruling the many trillions of humans in the Solar System, most of whom live aboard the Frame or in orbital habitats, an oligarchy has totalitarian control of many matters, including a deep secret about the actual disposition of the Sun. The crux of the drama is a quest by Muree, once she has become an ambassador from the starship to the Fortress Sol, as the Mask-bound society is called, to learn what is really going on inside the Wrap. A young man named Rab is Muree's opposite number, and their collaboration is soon seen as a conspiracy by the rulers of Fortress Sol.

My grasp of political science is tenuous at best. Naturally, the unraveling of a thousand-year-old political system is not for the faint of heart, and I wouldn't have a clue how to advise Mr. Baxter about it. So I'll stick with intriguing ideas, and leave it to you to read the book for yourself, and get from it what you are equipped to consume.

The Frame is found to be analogous to the nested-Platonic-Solid model of the Solar System devised by Kepler, before he determined that planetary orbits are (almost) ellipses. I say "almost" because the influence of all the planets upon each other perturbs the orbits a bit. There is a reason the planets are spaced far apart!

In Kepler's model, the five Platonic solids each enclose a sphere that just touches each planar face, and each is enclosed in a sphere that just touches all its corners. From outermost inward, they are a cube (8 faces), tetrahedron (4 faces), dodecahedron (12 faces), icosahedron (20 faces), and octahedron (8 faces). The fit is not exact, but the thickness of the spherical shells takes up the slack between the nearest and farthest points in each orbit from the Sun. The outermost sphere contains the orbit of Saturn, and so forth inward.

The Frame is planar, with rings and spokes. It is a rotating structure that has "trolleys" which move along the rings and inward or outward along the spokes. To get from a station on the Frame to a planet, one takes a shuttlecraft. A few numbers were used, and the math is fascinating.

First, the rotation of the outermost ring of the Frame, some distance outside the aphelion of Uranus, is sufficient to produce a centripetal force (rotation-induced artificial gravity) of 1/3 G. In the past, I did calculations to find out the stresses in a large, rotating ring with a centripetal force of 1 G. It turns out, if the radius of the ring is 1 km, we don't have any material that can sustain the stress, except perhaps carbon nanotubes, but only if they are continuous, with no breaks. Thus, a thread of nanotubes with a structure similar to cotton thread (composed of entwined strands a few cm long) is much weaker and would pull apart at much lower stress. I want to know if this Solar-System-spanning rotating Frame is feasible, based on science so far known.

Rough analysis: I don't know where my notebook is from doing that analysis, so we'll indulge in a bit of hand-waving instead. I'll begin with the basis that carbon nanotubes would be barely adequate to hold together a rotating ring of 1 km radius, with 1 G of centripetal force at its periphery. Our basic equation for centripetal force is F = Mv²/r.

• Subproblem 1: How big can r be if the target F is 1/3 G? Since v = αr, where α is angular velocity (degrees per second, perhaps), and we assume M is always 1 (that is, 1 kg), F = α²r. Thus, at constant angular velocity, F is proportional to r, so the radius for 1/3 G is 1/3 the radius for 1 G.
• Subproblem 2: What is the radius of the outer ring? It must be outside the aphelion distance of Uranus, which is 20.073 AU or just over three billion km. I don't know how much the passage of Uranus close to the Frame would perturb it, but let's ignore that for the moment, and posit a radius of 3.1 billion km.
• SP 3: The 1 G rotating, 1-km-radius ring has a peripheral velocity of 99 m/s and rotates one full turn every 63.5 s. We want F to be 9.8/3 = 3.267, and r is now 3.1 billion km (or 3.1 trillion m), which will equal v². The square root of the resulting 10.13 trillion (in units of squared velocity) is 3.182 million m/s or 3,182 km/s. This is about 470 times the orbital speed of Uranus. Uranus completes an orbit of the Sun every 84 years. Any point on this ring must therefore orbit the Sun about every 65 days. 

If you leave a station orbiting above Uranus and want to catch up to a station on the Frame, it takes a bit of a push. If you accelerate at 1 G in your shuttle:  v = at, or t = v/a. 3,182,000/9.8 = almost 325,000 seconds, or 3.75 days. If instead, you can handle 3 G's for thirty hours, that will do it.

I'll leave it to the physics-inclined to figure the hoop stress in the ring. You can be sure it is thousands of times the ultimate breaking stress of a carbon nanotube. 

Anyway, I figured all this after the fact. The "invention" of the lumes and the astonishing Solar-System-scale engineering imagined in Fortress Sol yielded quite an arena for space-cowboy derring-do, as Muree and her allies try to uncover the big lie at the root of the political system. What a fun read!